Voting by alternating veto - two people select a policy

Question

Econ 521 - Week 8:

1. Voting by alternating veto - Two people select a policy that affects them both by alternately vetoing policies until only one remains. First person 1 vetoes a policy. If more than one policy remains, person 2 then vetoes a policy. If more than one policy still remains, person 1 then vetoes another policy. The process continues until only one policy has not been vetoed. Suppose there are three possible policies, X, Y , and Z. Person 1 prefers X to Y to Z, and person 2 prefers Z to Y to X.

(a) Model this situation as an extensive game with perfect information. Specify the components of the game.

(b) Find its subgame perfect equilibria using backward induction.

(c) Find its Nash equilibria. Is there any Nash equilibria which is not a SPE? Is there any SPE which is not a NE?

2. A synergistic relationship - Consider a situation in which two (i = 1, 2) individuals are involved in a synergistic relationship. Suppose that the players choose their effort levels a_i that is a nonnegative number. Each individual preferences (for i = 1, 2) are represented by the payoff unction a_i(c + a_j - a_i), where j is the other individual and c > 0 is a constant. Based on this,

(a) If both individuals choose the effort levels simultaneously, find the Nash equilibrium.

(b) If individual 1 chooses first, find the SPE.

(c) Explain any difference between your answers in (a) and (b).

3. Removing stones - Two people take turns removing stones from a pile of n stones. Each person may, on each of her turns, remove either one stone or two stones. The person who takes the last stone is the winner; she gets $1 from her opponent. Find the subgame perfect equilibria of the games that model this situation for n = 1 and n = 2. Find the winner in each subgame perfect equilibrium for n = 3, using the fact that the subgame following player 1 removal of one stone is the game for n = 2 in which player 2 is the first-mover, and the subgame following player 1 removal of two stones is the game for n = 1 in which player 2 is the first mover. Use the same technique to find the winner in each SPE for n = 4.

4. Extensive game with simultaneous moves.

(a) Find all NE and SPE.